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The term biquad'' is short for bi-quadratic'', and is a common name for a two-pole, two-zero digital filter. The transfer function of the biquad can be defined as

 (11.8)

where is called the gain of the biquad. Since both the numerator and denominator of this transfer function are quadratic polynomials in , the transfer function is said to be bi-quadratic'' in .

As derived in §10.1.3, for real second-order polynomials having complex roots, it is often convenient to express the polynomial coefficients in terms of the radius and angle of the positive-frequency pole. For example, denoting the denominator polynomial by , we have

This representation is most often used for the denominator of the biquad, and we think of as the resonance frequency (in radians per sample-- , where is the resonance frequency in Hz), and determines the Q'' of the resonance (see §10.1.3). The numerator is less often represented in this way, but when it is, we may think of the zero-angle as the antiresonance frequency, and the zero-radius affects the depth and width of the antiresonance (or notch).

As discussed in §10.6.4, a common setting for the zeros when making a resonator is to place one at (dc) and the other at (half the sampling rate), i.e., and in Eq. (10.8) above . This zero placement normalizes the peak gain of the resonator if it is swept using the parameter.

Using the shift theorem for z transforms, the difference equation for the biquad can be written by inspection of the transfer function as

where denotes the input signal sample at time , and is the output signal. This is the form that is typically implemented in software. It is essentially the direct-form I implementation. (To obtain the official direct-form I structure, the overall gain must be not be pulled out separately, resulting in feedforward coefficients instead. See Chapter 9 for more about filter implementation forms.)

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